Abstract: Following a work of P. Souplet which presents an example showing the nonuniqueness of antiperiodic solutions of a second order ordinary equation, we show, using a method of W.S. Loud the existence of 4 antiperiodic solutions of the equation x” + cx’ + αx + βx3 = εƒ(t) for a function ƒ. Then we give concrete conditions for the existence of 3 or 4 periodic solutions of the same equation. In both cases ƒ can be chosen analytic.